Georgian Mathematical Journal
Volume 10 (2003), Number 1, 157–164
LOCAL GROWTH OF WEIERSTRASS σ-FUNCTION AND
WHITTAKER-TYPE DERIVATIVE SAMPLING
TIBOR K. POGÁNY
Dedicated to N. N. Leonenko on the occasion of his 50th birthday
Abstract. Two explicit guard functions Kj = Kj (δz ), j = 1, 2, are obtained,
which depend on the distance δz between z and the nearest point of the
2
integer lattice in the complex plane, such that δz K1 (δz ) ≤ |σ(z)|e−π|z| /2 ≤
δz K2 (δz ), z ∈ C, where σ(z) stands for the Weierstraß σ-function. This
result is used to improve the circular truncation error upper bound in the qth order Whittaker-type derivative sampling for the Leont’ev functions space
[2, πq
2 ), q ≥ 1.
2000 Mathematics Subject Classification: 30D15, 30E10, 94A20; Secondary: 30A10, 41A05, 94A12.
Key words and phrases: Circular truncation error, derivative sampling,
entire function spaces [ρ, ψ], [ρ, ψ), sampling truncation error upper bound,
Weierstraß σ-function, Whittaker-type derivative sampling reconstruction.
1. Introduction
The Weierstraß σ-function described as an infinite product is given by
¶
½
¾
Y0 µ
z
z
z2
σ(z) = z
1−
exp
+
,
m + ni
m + ni 2(m + ni)2
2
(m,n)∈Z
where the dashed product means that the factor with m = n = 0 is omitted.
The local growth estimation of σ(z), interesting for different purposes, has quite
a long history. The first result of this kind known by the author is given in [3],
where ln Mσ (r) ∼ πr2 /2, r → ∞, is proved (Mσ (r) stands for the maximum
modulus of σ on the circle |z| = r), see also [7, Chapter 4, §1, Problem 49],
where this result is quoted from the book [3]. After that Hayman proved that
there exist absolute constants K1 , K2 , K1 < K2 , for which
K1 ≤
|σ(z)|
2
e−π|z| /2 ≤ K2 ,
2
dist(z, Z )
z ∈ C.
(1)
In his article [1] no specific comments were given on the nature of Kj ’s. (Here
we have to point out that Hayman’s proof is not correct; he wrongly deduced
that the type of σ(z) is p
π/4 instead of the true value π/2, [1].) Seip confirms
(1) on the lattice Λα = απ Z2 , α > 0, but without any closer specification of
Kj ’s, [8] (these estimates of σ(z) were needed as convergence tools).
In this short note we obtain the values for these uniform constants (K1 ≈
0.266 and K2 = 1) as a corollary of our main result. Our principal result is to
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
158
TIBOR K. POGÁNY
establish two positive guard functions Kj = Kj (δz ), j = 1, 2, depending just
on δz = dist(z, Z2 ) such that (1) holds below; the proposed numerical bounds
(see the Proof of the Corollary 1), are the minimal and maximal values of these
guard functions respectively (therefore Kj cannot be improved in our setting).
In the third section of the article we consider the truncation error upper bound
appearing in the Whittaker-type derivative sampling restoration formula, i.e.
the sampling restoration formula which involves not just the sampled values of
the function f , but the sampled values of f (j) , j = 0, q − 1, i.e., of the first q
derivatives of the functions sampled at points of the lattice Z2 . This type of
sampling reconstruction holds for the Leont’ev type functions space [2, πq
) 1,
2
[4]. Our second main goal is to improve the truncation error upper bound in
[6], Theorem 1, with the aid of inequalities (2).
2. Local Growth Behaviour of σ(z)-Function
Along with the notation already introduced the following symbols will also
be used: N, Z, C denote the sets of natural, integer and complex numbers,
respectively; the symbol Cl{A} denotes the closure of the set A, t? is the complex
conjugate of t ∈ C, the circle Γr = {ζ| |ζ| = r} contains no point of Z2 .
Theorem 1. For all z ∈ C we have
n π
o
δz K1 (δz ) ≤ |σ(z)| exp − |z|2 ≤ δz K2 (δz ),
2
where
and A22 :=
µ
P
¶
n π o
¢2
π 4 δz4 ¡
K1 (δz ) = 1 −
1 − A22 δz4 exp − δz2 ,
90
2
½µ 4
¾
¶
π
π
K2 (δz ) = exp
+ A22 δz4 − δz2 ,
90
2
(2)
2
m,n∈N (m
(3)
(4)
+ n2 )−2 .
Proof. Let P 1 = (− 12 , 12 )2 be the period cell of the Weierstraß σ-function. First
2
we remark that for all z ∈ Cl{P 1 },
2
¶
½µ 4
¶ ¾
µ
4 4
π
π δz
4 2
≤ |σ(z)| ≤ δz exp
+ A22 δz4 .
δz (1 − A22 δz ) 1 −
(5)
90
90
Indeed, if z = x + iy ∈ Cl{P 1 }, then we conclude that δz = dist(z, Z2 ) =
2
dist(z, 0) = |z|. Put
¾
µ
¶
½
z2
z
z
+
.
am,n := 1 −
exp
m + ni
m + ni 2(m + ni)2
1The
infrequently used notation [ρ, ψ), [ρ, ψ] stands for the functions space consisting of
all entire functions of order less then or equal to ρ, and [ρ, ψ) denotes the case when the
function of order ρ has a type less then ψ, while [ρ, ψ] is used in the case when the function
of order ρ possesses a type less then or equal to ψ, [4].
LOCAL GROWTH OF WEIERSTRASS σ-FUNCTION
159
Consequently, we directly get
Y
Y
|σ(z)|2 = |z|2
|an,0 a−n,0 a0,n a0,−n |2
|am,n am,−n a−m,n a−m,−n |2
n∈N
m,n∈N
¶µ
¶¯2
Y ¯¯µ
z2
z 2 ¯¯
2
¯
= |z|
1+ 2 ¯
¯ 1 − n2
n
n∈N
¯Ã
!
µ
¶2 Ã
¶2 !
µ
Y ¯¯
z
z
×
1−
¯ 1−
¯
m
+
ni
m − ni
m,n∈N
½ µ
¶¾¯2
¯
1
1
2
¯
× exp z
+
¯
(m + ni)2 (m − ni)2
¶2
Yµ
|z|4
2
≤ |z|
1+ 4
n
n∈N
µ
¶
Y
(x2 − y 2 )(m2 − n2 ) + 4mnxy
|z|4
×
1−2
+
(m2 + n2 )2
(m2 + n2 )2
m,n∈N
µ
¶
(x2 − y 2 )(m2 − n2 ) − 4mnxy
|z|4
× 1−2
+
(m2 + n2 )2
(m2 + n2 )2
½
¾
(x2 − y 2 )(m2 − n2 )
× exp 4
.
(m2 + n2 )2
(5a)
(5b)
(5c)
(5d)
Now, we use 1 + t ≤ et to majorize the factors in displays (5b, c, d). So, straightforward calculation results in
( Ã
!
)
4
X
1
π
|σ(z)|2 ≤ |z|2 exp 2
+
|z|4 ,
(6)
2
2
2
90 m,n∈N (m + n )
which is the asserted upper bound in (5).
The lower bound derivation will be realized in a somewhat different way. At
first an auxiliary inequality is established. Namely, let a ∈ C, |a| < 1. Then
there holds
|(1 − a)ea | ≥ 1 − |a|2 .
(7)
Indeed, after fixing a = |a|eiφ , |a|, we get by direct calculation
p
h(φ) := |(1 − a)ea | = e|a| cos φ 1 − 2|a| cos φ + |a|2
≥ min h(φ) = h(0) = (1 − |a|)e|a| .
φ∈[0,2π)
Finally, by et ≥ 1 + t we deduce estimate (7).
P
To continue, assume λn ∈ (0, 1), n ∈ N. Then if n∈N λn converges, there
holds
Y
X
(1 − λn ) ≥ 1 −
λn .
(8)
n∈N
n∈N
160
TIBOR K. POGÁNY
This result is a generalization of the inequality due to Weierstraß. (In fact, (8)
is a straightforward consequence of the inequality concerning the finite product/sum case, cf. (1) in [5, 3.2.37., p. 207]).
Now we are ready to establish the lower bound for |σ(z)|, z ∈ Cl{P 1 }. We
2
have
¯
¶¯
¯Y µ
z 4 ¯¯
¯
|σ(z)| = |z| ¯
1− 4 ¯
¯
n ¯
n∈N
¯
¯
¶
µ
¶
¯ Y µ
¯
2
z2
z2
z2
z
¯
(m+ni)2
(m−ni)2 ¯¯
1−
(9)
ׯ
e
1
−
e
2
2
¯
¯
(m
+
ni)
(m
−
ni)
m,n∈N
¶ Y µ
¶2
Yµ
|z|4
|z|4
≥ |z|
1− 4
1−
(10)
2 + n2 )2
n
(m
n∈N
m,n∈N
Ã
!Ã
!2
X 1
X
1
≥ |z| 1 − |z|4
1 − |z|4
4
n
(m2 + n2 )2
n∈N
m,n∈N
µ
¶
¢2
π4 4 ¡
= |z| 1 − δz 1 − A22 δz4 .
(11)
90
Indeed, since for a = z 2 (m±in)−2 we have |a| ≤ 14 for all z ∈ Cl{P 1 }, m, n ∈ N,
2
this allows us to use estimate (7) in the double-indexed product in (9), and
(2)
(1)
then, taking into account that the factors λn = |z|4 n−4 , λn = |z|4 (m2 + n2 )−2
belong to the interval (0, 1) in display (10), with the help of the Weierstraß-type
inequality (8) we easily deduce (11).
Finally, combining estimates (6) and (11), we finish the derivation of (5).
Let τk : C 7→ Cl{P 1 } be the translation which defines a unique k = (ku , kv ) ∈
2
2
Z and a unique w = (u, v) ∈ Cl{P 1 } such that
2
z = k + w = ku + ikv + u + iv.
The quasi-periodicity property of the Weierstraß σ-function implies that
σ(z) = (−1)ku +kv +ku kv σ(w) exp{πwk? + π|k|2 /2}.
Since dist(z, Z2 ) = dist(w, 0) = δz , we get
|σ(z)| = |σ(w)|| exp{πwk? + π|k|2 /2}|
¶
¾
½µ 4
o
n π
π
π
4
2
2
+A22 δz + (|k| + 2ku u + 2kv v + |w| ) exp − |w|2
≤ δz exp
90
2
2
½µ 4
¶
¾
n
o
π
π
π 2
= δz exp
+ A22 δz4 − δz2 exp
|z| .
90
2
2
This finishes the proof of the upper bound assertion in (2). Finally it remains
to derive the lower bound in (2). For this we can argue in a similar way so that
by the left-hand estimate in (5) we deduce
|σ(z)| = |σ(w)|| exp{πwk? + π|k|2 /2}|
LOCAL GROWTH OF WEIERSTRASS σ-FUNCTION
161
¶
4
µ
nπ
o
n π o
¢2
π 4 δz ¡
1 − A22 δz4 exp − δz2 exp
|z|2 ,
90
2
2
which is the asserted lower bound.
¤
P
Remark 1. The Mathematica 4.0 gives A22 = m,n∈N (m2 +n2 )−2 ≈ 0.42437977.
So we can rearrange (3) and (4) with the approximate value of A22 .
≥ δz 1 −
Corollary 1.1. The following estimates hold:
³
n πo
n π
o
´¡
¢
2
A22 2
π4
δz 1 − 360 1 − 4
exp −
≤ |σ(z)| exp − |z| ≤ δz ,
4
2
z ∈ C.
√
Proof. We find min K1 (δz ) and max K2 (δz ) by (3), (4) as 0 ≤ δz ≤ 1/ 2. So
³
´¡
√
¢2 π
π4
K1 ⇐ min
K
(δ
)
=
K
(1/
2)
=
1
−
1 − A422 e− 4 ≈ 0, 26574548,
1 z
1
√
360
[0,1/ 2]
K2 ⇐ max
√ K2 (δz ) = K2 (0) = 1
[0,1/ 2]
are the uniform Hayman’s constants.
¤
Remark 2. The sharpness of inequalities (2) is an open question. Namely,
starting with (9) and (5a), by the Euler’s infinite product representation of
sin πδz and sinh πδz we get the lower and upper guard functions for |σ(z)| as
follows:
µ
¶2
n π o
sin
πδ
z
4 2
e
K1 (δz ) =
(1 − A22 δz ) exp − δz2 ,
πδz
4
µ
¶2
o
n
e 2 (δz ) = sinh πδz exp A22 δz4 − π δz2 .
K
πδz
4
e 1 (δz ) possesses lower minimum than K1 (δz ), and K
e 2 (δz ) has larger maxSince K
imum than K2 (δz ) we propose the use of (3) and (4) respectively.
3. Whittaker Type Derivative Sampling
In this part of the article we consider the influence of bounds (2) on the
convergence rate in the Whittaker-type q-th order derivative, uniformly spaced
sampling restoration formula for the Leont’ev function space [2, πq
).
2
πq
For f ∈ [2, 2 ϑ], ϑ ∈ [0, 1) we have
q
f (z) = σ (z)
q−1 q−1−j
X X
X
(m,n)∈Z2
j=0
k=0
q
f (q−1−j−k) (m + ni)Rmnj
j!(q − 1 − j − k)!(z − m − ni)k+1
(12)
uniformly on the compact subsets of C; as usual, f (0) (·) ≡ f (·). Here
µ
¶q
w − m − ni
dj
q
;
Rmnj = lim
w→m+ni dw j
σ(w)
cf. [6], Theorem 1 and Corollary 1 for certain additional details, e.g. truncation
error analysis, etc. (In fact, formula (12) for q = 1 belongs to Whittaker; the
case q > 1 appears in [2] as well.)
162
TIBOR K. POGÁNY
Let us introduce a subset of Z2 :
N(r) := {(m, n)| |m + ni| < r}.
Under the Whittaker-type derivative sampling series (12) truncated to N(r) we
mean the interpolation formula
IN (z; f ; σ; q; r) =
X
q−1 q−1−j
X
X
(m,n)∈N(r) j=0
k=0
q
σ q (z)f (q−1−j−k) (m + ni)Rmnj
; (13)
j!(q − 1 − j − k)!(z − m − ni))k+1
the so-called circular truncation error is
²N (f ; z; q; r) := f (z) − IN (z; f ; σ; q; r).
(14)
Now, we apply estimates (2) to the results in [6], Theorem 1, Corollary 1,
where we cannot avoid truncation error bound estimates such that depend on
the Hayman constants K1 , K2 . To prove the principal result in this section, we
will need the following evaluation.
Lemma 1. Consider a circle Γr = {ζ| |ζ| = r} such that contains no point
of Z2 . Then there holds
1 − |1 − 2(r2 − [r2 ])|
√
:= H(r),
4r+ 2
where [x] is the largest integer less than or equal to x.
δζ ≥ dist(Γr , Z2 ) ≥
(15)
Proof. Let A(a1 , a2 ) be the closest point in Z2 to Γr andpdenote by ∆A the
distance between A and the nearest point on Γr . Clearly, a21 + a22 = r ± ∆A ,
choosing the sign
or inside the circle Γr . Thus
p according to where A lies, outside
√
we can write a21 + a22 ≤ r + ∆A ≤ r + 1/ 2. Since a21 + a22 is the positive
integer nearest to r2 , we have
¡
¢
a21 + a22 = [r2 ] if A lies inside of Γr
= [r2 ] + 1 if A lies outside of Γr ,
which gives, that
q
¯
¯ |r2 − (a2 + a2 )|
¯
¯
2
p1
∆A = dist(Γr , Z2 ) = ¯r − a21 + a22 ¯ =
2
r + a1 + a22
min{r2 − [r2 ], [r2 ] + 1 − r2 }
√
≥
.
2r + 1/ 2
According to the definition of a modulus this implies (15).
¤
Theorem 2. For all f ∈ [2, πq
ϑ], ϑ ∈ [0, 1), and for all
2
"
#
1
|z|
+
N≥ p
+ 1, z ∈ C,
2(1 − ϑ) 2
we have
√
Af (2N + 1)(4N + 3)q e−2πq(1−ϑ)N
√
√
|²N (f ; z; q; 2(N + 1/2))| ≤ q
,
K1 (2(1 − 1 − ϑ)N + 1 + 1 − ϑ)
(16)
LOCAL GROWTH OF WEIERSTRASS σ-FUNCTION
163
where Af is the absolute constant which characterizes the [2, πq
ϑ]-function f ,
2
©
ª
2
i.e. it is given by |f (z)| ≤ Af exp πqϑ
|z|
,
z
∈
C.
Moreover
2
lim IN (z; f ; σ; q; r) = f (z)
N →∞
uniformly in z ∈ C.
Proof. First we repeat the procedure for deriving the truncation error upper
bound in [6], Theorem 1 with the integration path Γr = {ζ| |ζ| = r} chosen
according to the definition of the
© πqindex
ª set N(r) and assume z ∈ int(Γr ). We
2
point out that |f (z)| ≤ Af exp 2 ϑr on the circle Γr with Af > 0. Then
using (2), the numerical values of the Hayman constants K1 , K2 and (15) in
Lemma 1 for estimating |σ(·)| in this result in [6], we get
I
|σ(z)|q
|f (ζ)||dζ|
|²N (f ; z; q; r)| ≤
q
2π
Γr |σ(ζ)| |ζ − z|
µ
¶q
n πq
o
δz K2 (δz )
Af r
≤
exp
(|z|2 − (1 − ϑ)r2 )
minζ∈Γr δζ K1 (δζ ) r − |z|
2
µ
¶q
n
o
δz K2
Af r
πq
≤
exp
(|z|2 − (1 − ϑ)r2 )
H(r)K1
r − |z|
2
√ q
ª
© πq
Af r(4r + 2) exp 2 (|z|2 − (1 − ϑ)r2 )
(17)
≤
¡√
¢q .
(r − |z|) 2K1 (1 − |1 − 2(r2 − br2 c)|
√
It is not difficult to see that the circle Γr , r = 2(N + 1/2), does not contain
any integer point from Z2 , being r2 ∈
/ N. So Γ√2(N +1/2) is a suitable integration
√
contour for which (17) holds. Then, substituting r = 2(N + 1/2) into the
bound (17), we deduce
√
|²N (f ; z; q; 2(N + 1/2))|
Af (N + 1/2)(4(N + 1/2) + 1)q
2
2
√
eπq(1−ϑ)[(N −1/2) −(N +1/2) ]
q
K1 ((N + 1/2 − 1 − ϑ(N − 1/2))
Af (2N + 1)(4N + 3)q exp{−2πq(1 − ϑ)N }
√
√
=
,
Kq1 (2(1 − 1 − ϑ)N + 1 + 1 − ϑ)
≤
(18)
which is the asserted upper bound (16).
The uniform convergence in f (z) ≈ IN (z; f ; σ; q) follows from the truncation
error upper bound (16) as N → ∞. Indeed, since the right-hand term in
(16) does not depend on z and vanishes with the growth of N , the assertion
follows.
¤
Remark 3. By fixing the values of z we get the mathematical model
Af (2N + 1)(4N + 3)q e−2πq(1−ϑ)N
√
√
<²
Kq1 (2(1 − 1 − ϑ)N + 1 + 1 − ϑ)
(19)
for a pre-assigned approximation error level ² > 0 from (16). Then inequality
(19) gives an optimal value of N in finding the minimal size of the approximation
164
TIBOR K. POGÁNY
sum (13). Moreover, the convergence rate in (12) is
|²N (f ; z)| = O(N q e−2πq(1−ϑ)N )
under the assumptions of Theorem 2.
Acknowledgement
It is the author’s pleasure to thank Professor Emeritus J. R. Higgins for the
suggestions he made and for many helpful discussions. Many thanks are also
due to the unknown referee for detailed reading and making numerous valuable
comments.
References
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method. Canad. Math. Bull. 17(3)(1974), 317–358.
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155–171.
3. A. Hurwitz and R. Courant, Allgemeine Funktionentheorie und elliptische Funktionen. Geometrische Funktionentheorie. J. Springer, Berlin, 1922.
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5. D. S. Mitrinović, Analytic inequalities. (Serbian) Grad̄evinska knjiga, Beograd, 1970.
6. T. K. Pogány, Derivative uniform sampling via Weierstraß σ(z). Truncation error analπq
ysis in [2, 2s
2 ). Georgian Math. J. 8(2001), No. 1, 129–134.
7. G. Pólya and G. Szegő, Aufgaben und Lehrsätze aus der Analysis. Band II: Funktionentheorie. Nullstellen. Polynome. Determinanten. Zahlentheorie. Dritte berichtigte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 20, Springer-Verlag,
Berlin-New York, 1964 (Russian translation: Nauka, Moscow, 1978).
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Bargmann–Fock space II. J. Reine Angew. Math. 429(1992), 107–113.
(Received 2.07.2001; revised 21.09.2002)
Author’s address:
Faculty of Maritime Studies
University of Rijeka
51000 Rijeka, Studentska 2
Croatia
E-mail: poganj@brod.pfri.hr